propagating the arithmetics library, partial commit

[helm.git] / matita / matita / contribs / lambdadelta / ground / relocation / rtmap_after_nat_uni.ma

diff --git a/matita/matita/contribs/lambdadelta/ground/relocation/rtmap_after_nat_uni.ma b/matita/matita/contribs/lambdadelta/ground/relocation/rtmap_after_nat_uni.ma
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "ground/relocation/rtmap_uni.ma".
+include "ground/relocation/rtmap_nat.ma".
+include "ground/relocation/rtmap_after.ma".
+
+(* RELOCATION MAP ***********************************************************)
+
+lemma after_uni_dx: ∀l2,l1,f2. @↑❪l1, f2❫ ≘ l2 →
+                    ∀f. f2 ⊚ 𝐔❨l1❩ ≘ f → 𝐔❨l2❩ ⊚ ⫱*[l2] f2 ≘ f.
+#l2 @(nat_ind_succ … l2) -l2
+[ #l1 #f2 #Hf2 #f #Hf
+  elim (rm_nat_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
+  lapply (after_isid_inv_dx … Hf ?) -Hf
+  /3 width=3 by after_isid_sn, after_eq_repl_back0/
+| #l2 #IH #l1 #f2 #Hf2 #f #Hf
+  elim (rm_nat_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
+  [ #g2 #k1 #Hg2 #H1 #H2 destruct
+    elim (after_inv_pnx … Hf) -Hf [ |*: // ] #g #Hg #H destruct
+    <tls_xn /3 width=5 by after_next/
+  | #g2 #Hg2 #H2 destruct
+    elim (after_inv_nxx … Hf) -Hf [ |*: // ] #g #Hg #H destruct
+    <tls_xn /3 width=5 by after_next/
+  ]
+]
+qed.
+
+lemma after_uni_sn: ∀l2,l1,f2. @↑❪l1, f2❫ ≘ l2 →
+                    ∀f. 𝐔❨l2❩ ⊚ ⫱*[l2] f2 ≘ f → f2 ⊚ 𝐔❨l1❩ ≘ f.
+#l2 @(nat_ind_succ … l2) -l2
+[ #l1 #f2 #Hf2 #f #Hf
+  elim (rm_nat_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
+  lapply (after_isid_inv_sn … Hf ?) -Hf
+  /3 width=3 by after_isid_dx, after_eq_repl_back0/
+| #l2 #IH #l1 #f2 #Hf2 #f #Hf
+  elim (after_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H destruct
+  elim (rm_nat_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
+  [ #g2 #k1 #Hg2 #H1 #H2 destruct /3 width=7 by after_push/
+  | #g2 #Hg2 #H2 destruct /3 width=5 by after_next/
+  ]
+]
+qed-.
+